Nonlinear growth models represent an instance of nonlinear regression models, a class of models taking the general form \[ y = \mu(x, \theta) + \epsilon, \] where \(\mu(x, \theta)\) is the mean function which depends on a possibly vector-valued parameter \(\theta\), and a possibly vector-valued predictor \(x\). The stochastic component \(\epsilon\) represents the error with mean zero and constant variance. Usually, a Gaussian distribution is also assumed for the error term.
By defining the mean function \(\mu(x, \theta)\) we may obtain several different models, all characterized by the fact that parameters \(\theta\) enter in a nonlinear way into the equation. Parameters are usually estimated by nonlinear least squares which aims at minimizing the residual sum of squares.
\[ \mu(x) = \theta_1 \exp\{\theta_2 x\} \] where \(\theta_1\) is the value at the origin (i.e. \(\mu(x=0)\)), and \(\theta_2\) represents the (constant) relative ratio of change (i.e. \(\frac{d\mu(x)}{dx }\frac{1}{\mu(x)} = \theta_2\)). Thus, the model describes an increasing (exponential growth if \(\theta_2 > 0\)) or decreasing (exponential decay if \(\theta_2 < 0\)) trend with constant relative rate.
\[ \mu(x) = \frac{\theta_1}{1+\exp\{(\theta_2 - x)/\theta_3\}} \] where \(\theta_1\) is the upper horizontal asymptote, \(\theta_2\) represents the x-value at the inflection point of the symmetric growth curve, and \(\theta_3\) represents a scale parameter (and \(1/\theta_3\) is the growth-rate parameter that controls how quickly the curve approaches the upper asymptote).
\[ \mu(x) = \theta_1 \exp\{-\theta_2 \theta_3^x\} \] where \(\theta_1\) is the horizontal asymptote, \(\theta_2\) represents the value of the function at \(x = 0\) (displacement along the x-axis), and \(\theta_3\) represents a scale parameter.
The difference between the logistic and Gompertz functions is that the latter is not symmetric around the inflection point.
\[ \mu(x) = \theta_1 (1 - \exp\{-\theta_2 x\})^{\theta_3} \] where \(\theta_1\) is the horizontal asymptote, \(\theta_2\) represents the rate of growth, and \(\theta_3\) in part determines the point of inflection on the y-axis.
Dipartimento della Protezione Civile: COVID-19 Italia - Monitoraggio della situazione http://arcg.is/C1unv
Source: https://github.com/pcm-dpc/COVID-19
url = "https://raw.githubusercontent.com/pcm-dpc/COVID-19/master/dati-andamento-nazionale/dpc-covid19-ita-andamento-nazionale.csv"
COVID19 <- read.csv(file = url, stringsAsFactors = FALSE)
COVID19$data <- as.Date(COVID19$data)
DT::datatable(COVID19)Warnings
- 29/03/2020: dati Regione Emilia Romagna parziali (dato tampone non aggiornato).
- 26/03/2020: dati Regione Piemonte parziali (-50 deceduti - comunicazione tardiva)
- 18/03/2020: dati Regione Campania non pervenuti.
- 18/03/2020: dati Provincia di Parma non pervenuti.
- 17/03/2020: dati Provincia di Rimini non aggiornati
- 16/03/2020: dati P.A. Trento e Puglia non pervenuti.
- 11/03/2020: dati Regione Abruzzo non pervenuti.
- 10/03/2020: dati Regione Lombardia parziali.
- 07/03/2020: dati Brescia +300 esiti positivi
# create data for analysis
data = data.frame(date = COVID19$data,
y = COVID19$totale_casi)
data$x = as.numeric(data$date) - min(as.numeric(data$date)) + 1
DT::datatable(data, options = list("pageLength" = 5))mod1_start = lm(log(y) ~ x, data = data)
b = unname(coef(mod1_start))
start = list(th1 = exp(b[1]), th2 = b[2])
exponential <- function(x, th1, th2) th1 * exp(th2 * x)
mod1 = nls(y ~ exponential(x, th1, th2), data = data, start = start)
summary(mod1)
##
## Formula: y ~ exponential(x, th1, th2)
##
## Parameters:
## Estimate Std. Error t value Pr(>|t|)
## th1 4722.432751 638.646332 7.394 0.00000000724 ***
## th2 0.084196 0.003856 21.836 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 7738 on 38 degrees of freedom
##
## Number of iterations to convergence: 9
## Achieved convergence tolerance: 0.000008172mod2 = nls(y ~ SSlogis(x, Asym, xmid, scal), data = data)
summary(mod2)
##
## Formula: y ~ SSlogis(x, Asym, xmid, scal)
##
## Parameters:
## Estimate Std. Error t value Pr(>|t|)
## Asym 137178.03811 1416.73784 96.83 <2e-16 ***
## xmid 29.83619 0.15159 196.82 <2e-16 ***
## scal 5.65918 0.07438 76.08 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 886.2 on 37 degrees of freedom
##
## Number of iterations to convergence: 0
## Achieved convergence tolerance: 0.000004144mod3 = nls(y ~ SSgompertz(x, Asym, b2, b3), data = data)
# start = list(Asym = coef(mod2)[1])
# tmp = list(y = log(log(start$Asym) - log(data$y)), x = data$x)
# b = unname(coef(lm(y ~ x, data = tmp)))
# start = c(start, c(b2 = exp(b[1]), b3 = exp(b[2])))
# mod3 = nls(y ~ SSgompertz(x, Asym, b2, b3), data = data, start = start,
# control = nls.control(maxiter = 1000))
summary(mod3)
##
## Formula: y ~ SSgompertz(x, Asym, b2, b3)
##
## Parameters:
## Estimate Std. Error t value Pr(>|t|)
## Asym 203124.236720 6048.176421 33.58 <2e-16 ***
## b2 9.758029 0.312000 31.28 <2e-16 ***
## b3 0.929275 0.001875 495.48 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1039 on 37 degrees of freedom
##
## Number of iterations to convergence: 0
## Achieved convergence tolerance: 0.000001121richards <- function(x, th1, th2, th3) th1*(1 - exp(-th2*x))^th3
Loss <- function(th, y, x) sum((y - richards(x, th[1], th[2], th[3]))^2)
start <- optim(par = c(coef(mod2)[1], 0.001, 1), fn = Loss,
y = data$y, x = data$x)$par
names(start) <- c("th1", "th2", "th3")
mod4 = nls(y ~ richards(x, th1, th2, th3), data = data, start = start,
# trace = TRUE, algorithm = "plinear",
control = nls.control(maxiter = 1000, tol = 0.1))
# algorithm is not converging...
summary(mod4)
##
## Formula: y ~ richards(x, th1, th2, th3)
##
## Parameters:
## Estimate Std. Error t value Pr(>|t|)
## th1 231462.172564 13419.813759 17.25 <2e-16 ***
## th2 0.058588 0.003494 16.77 <2e-16 ***
## th3 6.404756 0.421883 15.18 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1321 on 37 degrees of freedom
##
## Number of iterations to convergence: 6
## Achieved convergence tolerance: 0.001708
# library(nlmrt)
# mod4 = nlxb(y ~ th1*(1 - exp(-th2*x))^th3,
# data = data, start = start, trace = TRUE)models = list("Exponential model" = mod1,
"Logistic model" = mod2,
"Gompertz model" = mod3,
"Richards model" = mod4)
tab = data.frame(loglik = sapply(models, logLik),
df = sapply(models, function(m) attr(logLik(m), "df")),
Rsquare = sapply(models, function(m)
cor(data$y, fitted(m))^2),
AIC = sapply(models, AIC),
AICc = sapply(models, AICc),
BIC = sapply(models, BIC))
sel <- apply(tab[,4:6], 2, which.min)
tab$"" <- sapply(tabulate(sel, nbins = length(models))+1, symnum,
cutpoints = 0:4, symbols = c("", "*", "**", "***"))
knitr::kable(tab)| loglik | df | Rsquare | AIC | AICc | BIC | ||
|---|---|---|---|---|---|---|---|
| Exponential model | -413.8861 | 3 | 0.9692804 | 833.7721 | 834.4388 | 838.8388 | |
| Logistic model | -326.6738 | 4 | 0.9996030 | 661.3477 | 662.4905 | 668.1032 | *** |
| Gompertz model | -333.0214 | 4 | 0.9994391 | 674.0428 | 675.1856 | 680.7983 | |
| Richards model | -342.6500 | 4 | 0.9991454 | 693.3001 | 694.4429 | 700.0556 |
ggplot(data, aes(x = date, y = y)) +
geom_point() +
geom_line(aes(y = fitted(mod1), color = "Exponential")) +
geom_line(aes(y = fitted(mod2), color = "Logistic")) +
geom_line(aes(y = fitted(mod3), color = "Gompertz")) +
geom_line(aes(y = fitted(mod4), color = "Richards")) +
labs(x = "", y = "Infected", color = "Model") +
scale_color_manual(values = cols) +
scale_y_continuous(breaks = seq(0, coef(mod2)[1], by = 5000),
minor_breaks = seq(0, coef(mod2)[1], by = 1000)) +
scale_x_date(date_breaks = "2 day", date_labels = "%b%d",
minor_breaks = "1 day") +
theme_bw() +
theme(legend.position = "top")last_plot() +
scale_y_continuous(trans = "log10", limits = c(100,NA)) +
labs(y = "Infected (log10 scale)")df = data.frame(x = seq(min(data$x), max(data$x)+14))
df = cbind(df, date = as.Date(df$x, origin = data$date[1]-1),
fit1 = predict(mod1, newdata = df),
fit2 = predict(mod2, newdata = df),
fit3 = predict(mod3, newdata = df),
fit4 = predict(mod4, newdata = df))
ylim = c(0, max(df[,c("fit2", "fit3")]))ggplot(data, aes(x = date, y = y)) +
geom_point() +
geom_line(data = df, aes(x = date, y = fit1, color = "Exponential")) +
geom_line(data = df, aes(x = date, y = fit2, color = "Logistic")) +
geom_line(data = df, aes(x = date, y = fit3, color = "Gompertz")) +
geom_line(data = df, aes(x = date, y = fit4, color = "Richards")) +
coord_cartesian(ylim = ylim) +
labs(x = "", y = "Infected", color = "Model") +
scale_y_continuous(breaks = seq(0, max(ylim), by = 10000),
minor_breaks = seq(0, max(ylim), by = 5000)) +
scale_x_date(date_breaks = "2 day", date_labels = "%b%d",
minor_breaks = "1 day") +
scale_color_manual(values = cols) +
theme_bw() +
theme(legend.position = "top",
axis.text.x = element_text(angle=60, hjust=1))# compute prediction using Moving Block Bootstrap (MBB) for nls
df = data.frame(x = seq(min(data$x), max(data$x)+14))
df = cbind(df, date = as.Date(df$x, origin = data$date[1]-1))
pred1 = cbind(df, "fit" = predict(mod1, newdata = df))
pred1[df$x > max(data$x), c("lwr", "upr")] = predictMBB.nls(mod1, df[df$x > max(data$x),])[,2:3]
pred2 = cbind(df, "fit" = predict(mod2, newdata = df))
pred2[df$x > max(data$x), c("lwr", "upr")] = predictMBB.nls(mod2, df[df$x > max(data$x),])[,2:3]
pred3 = cbind(df, "fit" = predict(mod3, newdata = df))
pred3[df$x > max(data$x), c("lwr", "upr")] = predictMBB.nls(mod3, df[df$x > max(data$x),])[,2:3]
pred4 = cbind(df, "fit" = predict(mod4, newdata = df))
pred4[df$x > max(data$x), c("lwr", "upr")] = predictMBB.nls(mod4, df[df$x > max(data$x),])[,2:3]
# predictions for next day
pred = rbind(subset(pred1, x == max(data$x)+1, select = 2:5),
subset(pred2, x == max(data$x)+1, select = 2:5),
subset(pred3, x == max(data$x)+1, select = 2:5),
subset(pred4, x == max(data$x)+1, select = 2:5))
print(pred, digits = 3)
## date fit lwr upr
## 41 2020-04-04 149062 127956 174001
## 411 2020-04-04 120429 118310 122619
## 412 2020-04-04 125407 122500 128435
## 413 2020-04-04 126049 122799 130020
ylim = c(0, max(pred2$upr, pred3$upr, na.rm=TRUE))ggplot(data, aes(x = date, y = y)) +
geom_point() +
geom_line(data = pred1, aes(x = date, y = fit, color = "Exponential")) +
geom_line(data = pred2, aes(x = date, y = fit, color = "Logistic")) +
geom_line(data = pred3, aes(x = date, y = fit, color = "Gompertz")) +
geom_line(data = pred4, aes(x = date, y = fit, color = "Richards")) +
geom_ribbon(data = pred1, aes(x = date, ymin = lwr, ymax = upr),
inherit.aes = FALSE, fill = cols[1], alpha=0.3) +
geom_ribbon(data = pred2, aes(x = date, ymin = lwr, ymax = upr),
inherit.aes = FALSE, fill = cols[2], alpha=0.3) +
geom_ribbon(data = pred3, aes(x = date, ymin = lwr, ymax = upr),
inherit.aes = FALSE, fill = cols[3], alpha=0.3) +
geom_ribbon(data = pred4, aes(x = date, ymin = lwr, ymax = upr),
inherit.aes = FALSE, fill = cols[4], alpha=0.3) +
coord_cartesian(ylim = c(0, max(ylim))) +
labs(x = "", y = "Infected", color = "Model") +
scale_y_continuous(minor_breaks = seq(0, max(ylim), by = 10000)) +
scale_x_date(date_breaks = "2 day", date_labels = "%b%d",
minor_breaks = "1 day") +
scale_color_manual(values = cols) +
theme_bw() +
theme(legend.position = "top",
axis.text.x = element_text(angle=60, hjust=1))# create data for analysis
data = data.frame(date = COVID19$data,
y = COVID19$deceduti)
data$x = as.numeric(data$date) - min(as.numeric(data$date)) + 1
DT::datatable(data, options = list("pageLength" = 5))mod1_start = lm(log(y) ~ x, data = data)
b = unname(coef(mod1_start))
start = list(th1 = exp(b[1]), th2 = b[2])
exponential <- function(x, th1, th2) th1 * exp(th2 * x)
mod1 = nls(y ~ exponential(x, th1, th2), data = data, start = start)
summary(mod1)
##
## Formula: y ~ exponential(x, th1, th2)
##
## Parameters:
## Estimate Std. Error t value Pr(>|t|)
## th1 284.865691 39.894980 7.14 0.0000000159 ***
## th2 0.101345 0.003902 25.97 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 728.9 on 38 degrees of freedom
##
## Number of iterations to convergence: 10
## Achieved convergence tolerance: 0.000002569mod2 = nls(y ~ SSlogis(x, Asym, xmid, scal), data = data)
summary(mod2)
##
## Formula: y ~ SSlogis(x, Asym, xmid, scal)
##
## Parameters:
## Estimate Std. Error t value Pr(>|t|)
## Asym 18317.85003 318.39809 57.53 <2e-16 ***
## xmid 32.91955 0.22299 147.62 <2e-16 ***
## scal 5.36369 0.09139 58.69 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 125 on 37 degrees of freedom
##
## Number of iterations to convergence: 0
## Achieved convergence tolerance: 0.000004974mod3 = nls(y ~ SSgompertz(x, Asym, b2, b3), data = data)
# manually set starting values
# start = list(Asym = coef(mod2)[1])
# tmp = list(y = log(log(start$Asym) - log(data$y)), x = data$x)
# b = unname(coef(lm(y ~ x, data = tmp)))
# start = c(start, c(b2 = exp(b[1]), b3 = exp(b[2])))
# mod3 = nls(y ~ SSgompertz(x, Asym, b2, b3), data = data, start = start,
# control = nls.control(maxiter = 10000))
summary(mod3)
##
## Formula: y ~ SSgompertz(x, Asym, b2, b3)
##
## Parameters:
## Estimate Std. Error t value Pr(>|t|)
## Asym 32799.114818 938.107425 34.96 <2e-16 ***
## b2 12.144088 0.277339 43.79 <2e-16 ***
## b3 0.934165 0.001288 725.55 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 73.06 on 37 degrees of freedom
##
## Number of iterations to convergence: 0
## Achieved convergence tolerance: 0.000001474richards <- function(x, th1, th2, th3) th1*(1 - exp(-th2*x))^th3
Loss <- function(th, y, x) sum((y - richards(x, th[1], th[2], th[3]))^2)
start <- optim(par = c(coef(mod2)[1], 0.001, 1), fn = Loss,
y = data$y, x = data$x)$par
names(start) <- c("th1", "th2", "th3")
mod4 = nls(y ~ richards(x, th1, th2, th3), data = data, start = start,
# trace = TRUE, algorithm = "port",
control = nls.control(maxiter = 1000))
summary(mod4)
##
## Formula: y ~ richards(x, th1, th2, th3)
##
## Parameters:
## Estimate Std. Error t value Pr(>|t|)
## th1 39629.087323 2233.929382 17.74 <2e-16 ***
## th2 0.053585 0.002365 22.66 <2e-16 ***
## th3 7.883054 0.369778 21.32 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 91.93 on 37 degrees of freedom
##
## Number of iterations to convergence: 3
## Achieved convergence tolerance: 0.00000342models = list("Exponential model" = mod1,
"Logistic model" = mod2,
"Gompertz model" = mod3,
"Richards model" = mod4)
tab = data.frame(loglik = sapply(models, logLik),
df = sapply(models, function(m) attr(logLik(m), "df")),
Rsquare = sapply(models, function(m)
cor(data$y, fitted(m))^2),
AIC = sapply(models, AIC),
AICc = sapply(models, AICc),
BIC = sapply(models, BIC))
sel <- apply(tab[,4:6], 2, which.min)
tab$"" <- sapply(tabulate(sel, nbins = length(models))+1, symnum,
cutpoints = 0:4, symbols = c("", "*", "**", "***"))
knitr::kable(tab)| loglik | df | Rsquare | AIC | AICc | BIC | ||
|---|---|---|---|---|---|---|---|
| Exponential model | -319.3914 | 3 | 0.9809653 | 644.7827 | 645.4494 | 649.8494 | |
| Logistic model | -248.3191 | 4 | 0.9994574 | 504.6382 | 505.7811 | 511.3937 | |
| Gompertz model | -226.8503 | 4 | 0.9997819 | 461.7007 | 462.8435 | 468.4562 | *** |
| Richards model | -236.0393 | 4 | 0.9996717 | 480.0787 | 481.2215 | 486.8342 |
ggplot(data, aes(x = date, y = y)) +
geom_point() +
geom_line(aes(y = fitted(mod1), color = "Exponential")) +
geom_line(aes(y = fitted(mod2), color = "Logistic")) +
geom_line(aes(y = fitted(mod3), color = "Gompertz")) +
geom_line(aes(y = fitted(mod4), color = "Richards")) +
labs(x = "", y = "Deceased", color = "Model") +
scale_color_manual(values = cols) +
scale_y_continuous(breaks = seq(0, coef(mod2)[1], by = 500),
minor_breaks = seq(0, coef(mod2)[1], by = 100)) +
scale_x_date(date_breaks = "2 day", date_labels = "%b%d",
minor_breaks = "1 day") +
theme_bw() +
theme(legend.position = "top")last_plot() +
scale_y_continuous(trans = "log10", limits = c(10,NA)) +
labs(y = "Deceased (log10 scale)")df = data.frame(x = seq(min(data$x), max(data$x)+14))
df = cbind(df, date = as.Date(df$x, origin = data$date[1]-1),
fit1 = predict(mod1, newdata = df),
fit2 = predict(mod2, newdata = df),
fit3 = predict(mod3, newdata = df),
fit4 = predict(mod4, newdata = df))
ylim = c(0, max(df[,-(1:3)]))ggplot(data, aes(x = date, y = y)) +
geom_point() +
geom_line(data = df, aes(x = date, y = fit1, color = "Exponential")) +
geom_line(data = df, aes(x = date, y = fit2, color = "Logistic")) +
geom_line(data = df, aes(x = date, y = fit3, color = "Gompertz")) +
geom_line(data = df, aes(x = date, y = fit4, color = "Richards")) +
coord_cartesian(ylim = ylim) +
labs(x = "", y = "Deceased", color = "Model") +
scale_y_continuous(breaks = seq(0, max(ylim), by = 1000),
minor_breaks = seq(0, max(ylim), by = 1000)) +
scale_x_date(date_breaks = "2 day", date_labels = "%b%d",
minor_breaks = "1 day") +
scale_color_manual(values = cols) +
theme_bw() +
theme(legend.position = "top",
axis.text.x = element_text(angle=60, hjust=1))# compute prediction using Moving Block Bootstrap (MBB) for nls
df = data.frame(x = seq(min(data$x), max(data$x)+14))
df = cbind(df, date = as.Date(df$x, origin = data$date[1]-1))
pred1 = cbind(df, "fit" = predict(mod1, newdata = df))
pred1[df$x > max(data$x), c("lwr", "upr")] = predictMBB.nls(mod1, df[df$x > max(data$x),])[,2:3]
pred2 = cbind(df, "fit" = predict(mod2, newdata = df))
pred2[df$x > max(data$x), c("lwr", "upr")] = predictMBB.nls(mod2, df[df$x > max(data$x),])[,2:3]
pred3 = cbind(df, "fit" = predict(mod3, newdata = df))
pred3[df$x > max(data$x), c("lwr", "upr")] = predictMBB.nls(mod3, df[df$x > max(data$x),])[,2:3]
pred4 = cbind(df, "fit" = predict(mod4, newdata = df))
pred4[df$x > max(data$x), c("lwr", "upr")] = predictMBB.nls(mod4, df[df$x > max(data$x),])[,2:3]
# predictions for next day
pred = rbind(subset(pred1, x == max(data$x)+1, select = 2:5),
subset(pred2, x == max(data$x)+1, select = 2:5),
subset(pred3, x == max(data$x)+1, select = 2:5),
subset(pred4, x == max(data$x)+1, select = 2:5))
print(pred, digits = 3)
## date fit lwr upr
## 41 2020-04-04 18163 16051 20436
## 411 2020-04-04 14994 14577 15322
## 412 2020-04-04 15582 15380 15775
## 413 2020-04-04 15656 15401 15959
ylim = c(0, max(pred2$upr, pred3$upr, na.rm=TRUE))ggplot(data, aes(x = date, y = y)) +
geom_point() +
geom_line(data = pred1, aes(x = date, y = fit, color = "Exponential")) +
geom_line(data = pred2, aes(x = date, y = fit, color = "Logistic")) +
geom_line(data = pred3, aes(x = date, y = fit, color = "Gompertz")) +
geom_line(data = pred4, aes(x = date, y = fit, color = "Richards")) +
geom_ribbon(data = pred1, aes(x = date, ymin = lwr, ymax = upr),
inherit.aes = FALSE, fill = cols[1], alpha=0.3) +
geom_ribbon(data = pred2, aes(x = date, ymin = lwr, ymax = upr),
inherit.aes = FALSE, fill = cols[2], alpha=0.3) +
geom_ribbon(data = pred3, aes(x = date, ymin = lwr, ymax = upr),
inherit.aes = FALSE, fill = cols[3], alpha=0.3) +
geom_ribbon(data = pred4, aes(x = date, ymin = lwr, ymax = upr),
inherit.aes = FALSE, fill = cols[4], alpha=0.3) +
coord_cartesian(ylim = c(0, max(ylim))) +
labs(x = "", y = "Deceased", color = "Model") +
scale_y_continuous(minor_breaks = seq(0, max(ylim), by = 1000)) +
scale_x_date(date_breaks = "2 day", date_labels = "%b%d",
minor_breaks = "1 day") +
scale_color_manual(values = cols) +
theme_bw() +
theme(legend.position = "top",
axis.text.x = element_text(angle=60, hjust=1))# create data for analysis
data = data.frame(date = COVID19$data,
y = COVID19$dimessi_guariti)
data$x = as.numeric(data$date) - min(as.numeric(data$date)) + 1
DT::datatable(data, options = list("pageLength" = 5))mod1_start = lm(log(y) ~ x, data = data)
b = unname(coef(mod1_start))
start = list(th1 = exp(b[1]), th2 = b[2])
exponential <- function(x, th1, th2) th1 * exp(th2 * x)
mod1 = nls(y ~ exponential(x, th1, th2), data = data, start = start)
summary(mod1)
##
## Formula: y ~ exponential(x, th1, th2)
##
## Parameters:
## Estimate Std. Error t value Pr(>|t|)
## th1 334.398837 37.440406 8.931 0.0000000000716 ***
## th2 0.103708 0.003111 33.336 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 726.2 on 38 degrees of freedom
##
## Number of iterations to convergence: 10
## Achieved convergence tolerance: 0.00000452mod2 = nls(y ~ SSlogis(x, Asym, xmid, scal), data = data)
summary(mod2)
##
## Formula: y ~ SSlogis(x, Asym, xmid, scal)
##
## Parameters:
## Estimate Std. Error t value Pr(>|t|)
## Asym 28353.1201 1173.7517 24.16 <2e-16 ***
## xmid 35.4780 0.5399 65.71 <2e-16 ***
## scal 6.0837 0.1700 35.78 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 254.7 on 37 degrees of freedom
##
## Number of iterations to convergence: 0
## Achieved convergence tolerance: 0.000008046mod3 = nls(y ~ SSgompertz(x, Asym, b2, b3), data = data)
summary(mod3)
##
## Formula: y ~ SSgompertz(x, Asym, b2, b3)
##
## Parameters:
## Estimate Std. Error t value Pr(>|t|)
## Asym 73354.261476 6402.745119 11.46 0.0000000000000993 ***
## b2 9.745093 0.247150 39.43 < 2e-16 ***
## b3 0.951305 0.002065 460.64 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 162.2 on 37 degrees of freedom
##
## Number of iterations to convergence: 1
## Achieved convergence tolerance: 0.0000004446richards <- function(x, th1, th2, th3) th1*(1 - exp(-th2*x))^th3
Loss <- function(th, y, x) sum((y - richards(x, th[1], th[2], th[3]))^2)
start <- optim(par = c(coef(mod2)[1], 0.001, 1), fn = Loss,
y = data$y, x = data$x)$par
names(start) <- c("th1", "th2", "th3")
mod4 = nls(y ~ richards(x, th1, th2, th3), data = data, start = start,
# trace = TRUE, # algorithm = "port",
control = nls.control(maxiter = 1000))
summary(mod4)
##
## Formula: y ~ richards(x, th1, th2, th3)
##
## Parameters:
## Estimate Std. Error t value Pr(>|t|)
## th1 158650.383366 39513.368487 4.015 0.000279 ***
## th2 0.026429 0.003651 7.239 0.0000000137 ***
## th3 4.897151 0.320984 15.257 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 167.8 on 37 degrees of freedom
##
## Number of iterations to convergence: 20
## Achieved convergence tolerance: 0.000007053models = list("Exponential model" = mod1,
"Logistic model" = mod2,
"Gompertz model" = mod3,
"Richards model" = mod4)
tab = data.frame(loglik = sapply(models, logLik),
df = sapply(models, function(m) attr(logLik(m), "df")),
Rsquare = sapply(models, function(m)
cor(data$y, fitted(m))^2),
AIC = sapply(models, AIC),
AICc = sapply(models, AICc),
BIC = sapply(models, BIC))
sel <- apply(tab[,4:6], 2, which.min)
tab$"" <- sapply(tabulate(sel, nbins = length(models))+1, symnum,
cutpoints = 0:4, symbols = c("", "*", "**", "***"))
knitr::kable(tab)| loglik | df | Rsquare | AIC | AICc | BIC | ||
|---|---|---|---|---|---|---|---|
| Exponential model | -319.2420 | 3 | 0.9885727 | 644.4841 | 645.1507 | 649.5507 | |
| Logistic model | -276.7965 | 4 | 0.9984391 | 561.5931 | 562.7359 | 568.3486 | |
| Gompertz model | -258.7605 | 4 | 0.9993022 | 525.5211 | 526.6640 | 532.2766 | *** |
| Richards model | -260.1210 | 4 | 0.9992782 | 528.2421 | 529.3849 | 534.9976 |
ggplot(data, aes(x = date, y = y)) +
geom_point() +
geom_line(aes(y = fitted(mod1), color = "Exponential")) +
geom_line(aes(y = fitted(mod2), color = "Logistic")) +
geom_line(aes(y = fitted(mod3), color = "Gompertz")) +
geom_line(aes(y = fitted(mod4), color = "Richards")) +
labs(x = "", y = "Recovered", color = "Model") +
scale_color_manual(values = cols) +
scale_y_continuous(breaks = seq(0, coef(mod2)[1], by = 500),
minor_breaks = seq(0, coef(mod2)[1], by = 100)) +
scale_x_date(date_breaks = "2 day", date_labels = "%b%d",
minor_breaks = "1 day") +
theme_bw() +
theme(legend.position = "top")last_plot() +
scale_y_continuous(trans = "log10", limits = c(10,NA)) +
labs(y = "Recovered (log10 scale)")df = data.frame(x = seq(min(data$x), max(data$x)+14))
df = cbind(df, date = as.Date(df$x, origin = data$date[1]-1),
fit1 = predict(mod1, newdata = df),
fit2 = predict(mod2, newdata = df),
fit3 = predict(mod3, newdata = df),
fit4 = predict(mod4, newdata = df))
ylim = c(0, max(df[,-(1:3)]))ggplot(data, aes(x = date, y = y)) +
geom_point() +
geom_line(data = df, aes(x = date, y = fit1, color = "Exponential")) +
geom_line(data = df, aes(x = date, y = fit2, color = "Logistic")) +
geom_line(data = df, aes(x = date, y = fit3, color = "Gompertz")) +
geom_line(data = df, aes(x = date, y = fit4, color = "Richards")) +
coord_cartesian(ylim = ylim) +
labs(x = "", y = "Recovered", color = "Model") +
scale_y_continuous(breaks = seq(0, max(ylim), by = 1000),
minor_breaks = seq(0, max(ylim), by = 1000)) +
scale_x_date(date_breaks = "2 day", date_labels = "%b%d",
minor_breaks = "1 day") +
scale_color_manual(values = cols) +
theme_bw() +
theme(legend.position = "top",
axis.text.x = element_text(angle=60, hjust=1))# compute prediction using Moving Block Bootstrap (MBB) for nls
df = data.frame(x = seq(min(data$x), max(data$x)+14))
df = cbind(df, date = as.Date(df$x, origin = data$date[1]-1))
pred1 = cbind(df, "fit" = predict(mod1, newdata = df))
pred1[df$x > max(data$x), c("lwr", "upr")] = predictMBB.nls(mod1, df[df$x > max(data$x),])[,2:3]
pred2 = cbind(df, "fit" = predict(mod2, newdata = df))
pred2[df$x > max(data$x), c("lwr", "upr")] = predictMBB.nls(mod2, df[df$x > max(data$x),])[,2:3]
pred3 = cbind(df, "fit" = predict(mod3, newdata = df))
pred3[df$x > max(data$x), c("lwr", "upr")] = predictMBB.nls(mod3, df[df$x > max(data$x),])[,2:3]
pred4 = cbind(df, "fit" = predict(mod4, newdata = df))
pred4[df$x > max(data$x), c("lwr", "upr")] = predictMBB.nls(mod4, df[df$x > max(data$x),])[,2:3]
# predictions for next day
pred = rbind(subset(pred1, x == max(data$x)+1, select = 2:5),
subset(pred2, x == max(data$x)+1, select = 2:5),
subset(pred3, x == max(data$x)+1, select = 2:5),
subset(pred4, x == max(data$x)+1, select = 2:5))
print(pred, digits = 3)
## date fit lwr upr
## 41 2020-04-04 23491 21296 25939
## 411 2020-04-04 20202 19411 20916
## 412 2020-04-04 20836 20387 21324
## 413 2020-04-04 20987 20540 21524
ylim = c(0, max(pred2$upr, pred3$upr, na.rm=TRUE))ggplot(data, aes(x = date, y = y)) +
geom_point() +
geom_line(data = pred1, aes(x = date, y = fit, color = "Exponential")) +
geom_line(data = pred2, aes(x = date, y = fit, color = "Logistic")) +
geom_line(data = pred3, aes(x = date, y = fit, color = "Gompertz")) +
geom_line(data = pred4, aes(x = date, y = fit, color = "Richards")) +
geom_ribbon(data = pred1, aes(x = date, ymin = lwr, ymax = upr),
inherit.aes = FALSE, fill = cols[1], alpha=0.3) +
geom_ribbon(data = pred2, aes(x = date, ymin = lwr, ymax = upr),
inherit.aes = FALSE, fill = cols[2], alpha=0.3) +
geom_ribbon(data = pred3, aes(x = date, ymin = lwr, ymax = upr),
inherit.aes = FALSE, fill = cols[3], alpha=0.3) +
geom_ribbon(data = pred4, aes(x = date, ymin = lwr, ymax = upr),
inherit.aes = FALSE, fill = cols[4], alpha=0.3) +
coord_cartesian(ylim = c(0, max(ylim))) +
labs(x = "", y = "Recovered", color = "Model") +
scale_y_continuous(breaks = seq(0, max(ylim), by = 5000),
minor_breaks = seq(0, max(ylim), by = 1000)) +
scale_x_date(date_breaks = "2 day", date_labels = "%b%d",
minor_breaks = "1 day") +
scale_color_manual(values = cols) +
theme_bw() +
theme(legend.position = "top",
axis.text.x = element_text(angle=60, hjust=1))df = data.frame(date = COVID19$data,
positives = c(NA, diff(COVID19$totale_casi)),
swabs = c(NA, diff(COVID19$tamponi)))
df$x = as.numeric(df$date) - min(as.numeric(df$date)) + 1
df$y = df$positives/df$swabs
df = subset(df, swabs > 50)
DT::datatable(df[,-4], )ggplot(df, aes(x = date)) +
geom_point(aes(y = swabs, color = "swabs"), pch = 19) +
geom_line(aes(y = swabs, color = "swabs")) +
geom_point(aes(y = positives, color = "positives"), pch = 0) +
geom_line(aes(y = positives, color = "positives")) +
labs(x = "", y = "Number of cases", color = "") +
scale_x_date(date_breaks = "2 day", date_labels = "%b%d",
minor_breaks = "1 day") +
scale_color_manual(values = palette()[c(2,1)]) +
theme_bw() +
theme(legend.position = "top",
axis.text.x = element_text(angle=60, hjust=1))ggplot(df, aes(x = date, y = y)) +
geom_smooth(method = "loess", se = TRUE, col = "darkgrey") +
geom_point(col=palette()[4]) +
geom_line(size = 0.5, col=palette()[4]) +
labs(x = "", y = "Positives / Admnistered swabs") +
scale_y_continuous(labels = scales::percent_format()) +
scale_x_date(date_breaks = "2 day", date_labels = "%b%d",
minor_breaks = "1 day") +
theme_bw() +
theme(legend.position = "top",
axis.text.x = element_text(angle=60, hjust=1))